Find characteristic, minimal, principal vector and Frobenius form from Jordan form I am given a Jordan form:
$$\left(\begin{array}{rrrrrrr}
2 & 1 &  &  &  &  &  \\
 & 2 & 1 &  &  &  &  \\
 &  & 2 &  &  &  &  \\
 &  &  & 2 & 1 &  &  \\
 &  &  &  & 2   &  \\
 &  &  &  &  & 1 &  \\
 &  &  &  &  &  & 0 \\
\end{array}\right)$$
I am thinking the characteristic of this matrix is $p(\lambda) = (\lambda - 2)^5(\lambda -1)\lambda$. I am guessing the minimal form is $m_A(x) = x(x-2)^3(x-2)^2(x-1)$, and, I am unsure about it though. But, I don't know how to find the principal vector and Frobenius form of the Jordan form. Can you please explain for me?
 A: For the Frobenius form (also called Rational Canonical Form) you need to find the invariant factors (of the $K[X]$-module) associated to the matrix$~A$, the largest of which is the minimal polynomial. In general that can be done by finding the Smith normal form of $XI-A$ in the set of square matrices over $K[X]$, but since $A$ is already in Jordan form there is an easier method. The (global) minimal polynomial is the least common multiple of the minimal polynomials of all Jordan blocks. This means for every eigenvalue only the largest blocks contribute to the minimal polynomial.  For the remaining invariant factors (if any), which divide the minimal polynomial, remove those blocks that contributed to the minimal polynomial, and repeat until no blocks remain. For the sequence of polynomials so obtained (which are usually taken in reverse order, so the minimal polynomial is the last one), the Frobenius form is obtained as a block diagonal matrix with the companion matrices of the polynomials as blocks. In the example you got two such companion matrix blocks, one for $(X-2)^3(X-1)X=X^5-4X^4+15X^3-20X^2+8X$ (the minimal polynomial) and one for $(X-2)^2=X^2-4X+4$.
